# Hedge Ratio

A hedge ratio is the ratio of exposure to a hedging instrument to the value of the hedged asset. A ratio of 1 or 100% means that the position is fully hedged and a ratio of 0 means it is not hedged at all.

Hedge ratio is an important statistic in risk management because it tells us the extent to which the risk of any adverse movement in our asset or liability will be met by any offsetting movement in the hedging instrument. As you will see in the example below, the hedge ratio changes due to changes in value of the hedging instrument and/or the hedged asset or liability. There are two ways in which a hedge can be set up: (a) a static hedge or (b) a dynamic hedge. In a static hedge, the number of hedging contracts is not changed over the course of the hedge in response to any movement in the values of the hedging instrument or the hedged asset. In a dynamic hedge, on the other hand, more hedging contracts are bought or sold to bring back the hedge ratio to the target hedge ratio.

### Optimal Hedge Ratio

An optimal hedge ratio (also called minimum-variance hedge ratio) is a ratio that tells use the percentage of our asset or liability exposure that we should hedge. It equals the product of the correlation between the prices of the hedging instrument and the hedged instrument and the volatility of the hedged instrument divided by the volatility of the hedging instrument. An optimal hedge ratio is most relevant where the characteristics of the hedged instrument and the hedging instrument are different i.e. in a cross hedge.

## Formula

Hedge ratio equals the value of the hedging instrument divided by the value of the hedged asset. It can be calculated using the following formula:

$$Hedge\ Ratio=\frac{h}{U}=\frac{c_u-c_d}{U_u-U_d}$$

Where h is the exposure to the hedging instrument and U is the value of the underlying i.e. hedged asset. hu and hd represent the value of the hedging instrument (forward, option, etc.) when the value of the underlying (i.e. the hedged asset) goes up and down respectively. Similarly, Uu and Ud represent the value of the underlying asset (i.e. the hedged asset) in the up and down states.

## Example

Let’s say you have a portfolio of stocks valued at EUR 1,000,000 that you want to hold for three more months. In the meanwhile, you don’t want any foreign exchange movement between Euro and US Dollar to spoil your returns. There are many ways in which you can hedge your exposure to Euro including selling Euros forward, buying put option on Euro, etc. Let’s say you sell 1 million Euros forward at 1.13\$/€. At the start of the hedge, you have a hedge ratio of 1 (i.e. value of the forward position of EUR 1,000,000 to the value of underlying of EUR 1,000,000). It is quite likely that the value of your portfolio will increase or decrease during the three-month period. Any such movement will change your hedge ratio such that your Euro exposure will be underhedged or over-hedged. If your portfolio grows to EUR 1,025,000 at the end of first month, you hedge ratio will fall to 97.5% (also written as 0.975). You have two options in such a scenario: (a) do nothing or (b) adjust your hedge position to match the movement in portfolio. You do nothing if you have a static hedge. However, if your hedge is a dynamic hedge, you will sell 5,000 more euros two-month forward.

### Example: Optimal Hedge Ratio

Since the underlying in this case is a stock portfolio denominated in Euro, hedging it with currency futures represents a cross hedge owing to the mismatch between their characteristics, you need to work out the optimal hedge ratio and hedge only that proportion of your portfolio instead of the whole portfolio. If the volatility of your stock portfolio is 8%, the volatility of the Euro futures contract is 10% and the correlation between your portfolio and the future contract is 0.5, your optimal hedge ratio works out to 40%.

$$Optimal\ Hedge\ Ratio=\rho\times\frac{\sigma_p}{\sigma_h}=0.5\times\frac{8\%}{10\%}=40\%$$

It means that instead of hedging 100% of your portfolio, you should hedge only 40%.

Written by Obaidullah Jan, ACA, CFA and last revised on